Octoberfest 2023
The 2023 Category Theory Octoberfest took place on the weekend of October 28th through October 29th. The meeting was virtual.
VIDEOS AND SLIDES
(Titles and abstracts can be found below the videos and slides.)
SATURDAY, October 28th
-Ettore Aldrovandi VIDEO, SLIDES
-Brandon Shapiro VIDEO, SLIDES
-Ana Luiza Tenorio VIDEO, SLIDES
-Miloslav Štěpán VIDEO, SLIDES
-Udit Mavinkurve VIDEO, SLIDES
-Christian Williams VIDEO, SLIDES
SUNDAY, October 29th
-Nadja Egner SLIDES
-Kristof Kanalas VIDEO, SLIDES
-Amartya Shekhar Dubey VIDEO, SLIDES
TITLES
Jacob Neumann: Paranatural Category Theory
Abstract: In this talk, I’ll describe my work towards carving out a (novel?) branch of category theory, dubbed paranatural category theory. The central objects of study in paranatural CT are paranatural transformations (known in the literature as strong dinatural, or Barr dinatural, transformations), which are a kind of transformation between difunctors (functors of the form ℂop × ℂ → Set), intermediate between the standard category-theoretic notions of dinatural transformations and natural transformations. I’ll detail the basic theory of paranatural transformations and a di-variant analogue of presheaf toposes, including a lovely “diYoneda Lemma” and an accompanying calculus of structural (co)ends. I’ll also pose the (still open) question of whether these constructions can be viewed as instances of the usual theory of natural transformations. Time permitting, I will explore some of the exciting applications of this theory: a category-theoretic treatment of parametrically-polymorphic functional programming, impredicative encodings of (co)inductive types, representation independence of abstract data structures, and difunctor models of dependent type theory.
A preprint covering this material is available on the arXiv at arxiv.org/abs/2307.09289.
Davide Trotta: A characterization of regular and exact completions of existential completions
Abstract: The notions of regular and exact completions of elementary and existential doctrines were brought up in recent works by M. E. Maietti, F. Pasquali, and P. Rosolini, inspired by the tripos-to-topos construction by M. Hyland, P. Johnstone and A. Pitts.
In this talk, we provide a characterization of the regular and exact completions of doctrines arising as instances of a free construction called existential completion. In particular, we show that these amount to the reg/lex and ex/lex-completions, respectively, of the category of predicates of their generating elementary doctrines. This characterization generalizes a previous result obtained by M.E. Maietti, F. Pasquali, and P. Rosolini on doctrines equipped with Hilbert’s epsilon-operators.
To achieve this goal, we introduce in the language of doctrines the notion of “existential free element” and of “existential cover”, and we show that these notions play analogous roles in the regular and exact completion of a doctrine as regular projective and projective cover do in the reg/lex and ex/lex completion.
Relevant examples of applications of our characterization, quite different from those involving doctrines with Hilbert’s epsilon-operators, include the regular syntactic category of the regular fragments of first-order logic (and his effectivization) as well as the construction of Joyal’s Arithmetic Universes.
This talk is based on a joint work with Maria Emilia Maietti.
Ettore Aldrovandi: Determinant functors for triangulated categories and categorical rings
Abstract: Determinant functors are functors into categorical groups and are equipped with additive data that convert distinguished triangles into sums. They port to the triangulated world the better behaved notion of determinant for exact categories. Tensor triangulated categories have an extra monoidal structure of their own, and one expects the determinants to be compatible with that, as it the case for exact categories. This is indeed the case: we prove the universal determinant for a tensor triangulated category takes values in a categorical ring. The proof relies on the underlying multi-categorical nature of both sides.
Brandon Shapiro: Finite posets as algebraic expressions in duoidal categories
Abstract: Duoidal categories, namely categories with two monoidal structures equipped with lax interchange morphisms, model a wide variety of different structures in mathematics such as certain categories of endofunctors and tropical real numbers. In the case when the two monoidal structures share a unit and the first is coherently symmetric, I will describe how the algebraic structure of the duoidal category is controlled by “expressible” posets, which are built out of algebraic expressions using disjoint unions and joins. As an example, using the duoidal structure on the nonnegative real numbers given by max and +, finite posets of program dependencies can be used to compute optimal runtimes of parallel computations. Based on joint work with David Spivak.
Ana Luiza: Sheaves on rings and Cech Cohomology
Abstract: A presheaf on a topological space X is a functor from a poset category – where the objects are open subsets of X and the morphisms are reverse inclusions. Sheaves are presheaves that satisfy a certain gluing property, and, categorically, they can be described by an equalizer diagram. Two concepts are essential in sheaf’s definition: the covering for an open of X, as a union of smaller open subsets, and the intersections between the open subsets that are part of the covering. It is well known that we can replace the union by the join and the intersection by the meet to obtain sheaves on locales (complete lattices in which finite meets distribute over arbitrary joins)
In this talk we will work with sheaves on quantales, a generalization of locales in which a monoidal operation distributes over arbitrary joins. In particular, we are interested in the quantale of ideals of a commutative ring with unity. We define a Cech cohomology of such a ring inspired by the definition for topological spaces but instead of the intersection of open subsets we have a multiplication of ideals. If C(X) is the ring of continuous real-valued functions on X, then the Cech cohomology groups of X are isomorphic to the Cech cohomology groups of C(X).
Miloslav Štěpán: Quasi-limits and lax flexibility
Abstract: In this talk I will discuss lax analogues of some concepts from two-dimensional monad theory.
I will recall the notion of a quasi-limit [1] – a lax analogue of a bilimit
where the equivalence is replaced by an adjunction. I will demonstrate that
these things appear in nature: the Kleisli 2-category for a lax-idempotent
2-monad (2-comonad) is complete (cocomplete) in this sense. This covers
examples such as the 2-category of strict monoidal categories and lax monoidal functors, or the bicategory Prof of categories and profunctors.
If time permits, I will remind the listener of the notion of a flexible
algebra for a 2-monad [2] and introduce its lax analogue. I will show that for
a special class of 2-monads, these “lax-flexible” algebras are equivalent to
generalized multicategories. For instance, lax-flexible strict monoidal
category is a free monoidal category on some underlying multicategory.
[1] Formal category theory: adjointness for 2-categories
[2] Two-dimensional monad theory
Udit Mavinkurve: The fundamental groupoid in discrete homotopy theory
Abstract: Discrete homotopy theory is a homotopy theory designed for studying simple graphs, detecting combinatorial, rather than topological, “holes.” Central to this theory are the discrete homotopy groups, defined using maps out of grids of suitable dimensions. Of these, the discrete fundamental group in particular has found applications in various areas of mathematics, including matroid theory, hyperplane arrangements, and topological data analysis.
In this talk, based on joint work with C. Kapulkin (arxiv:2303.06029), we introduce the discrete fundamental groupoid and use it as a starting point to develop some robust computational techniques. A new notion of covering graphs allows us to extend the existing theory of universal covers to all graphs, and to prove a classification theorem for coverings. We also prove a discrete version of the Seifert-van Kampen theorem, generalizing a previous result of H. Barcelo et al. We then use it to solve the realization problem for the fundamental groupoid through a purely combinatorial construction.
Jack Romo: Constructing Homotopy Bicategories of Complete 2-fold Segal Spaces
Abstract: Across the multitude of definitions for a higher category, a dividing line can be found between two major camps of model. On one side lives the ‘algebraic’ models, like Bénabou’s bicategories, tricategories following Gurski and the models of Batanin and Leinster, Trimble and Penon. On the other end, one finds the ‘non-algebraic’ models, including those of Tamsamani and Paoli, along with quasicategories, Segal n-categories, complete n-fold Segal spaces and more. The bridges between these models remain somewhat mysterious. Progress has been made in certain instances, as seen in the work of Tamsamani, Leinster, Lack and Paoli, Cottrell, Campbell and others. Nonetheless, the correspondence remains incomplete; indeed, for instance, there is no fully verified means in the literature to take an `algebraic’ homotopy n-category of any known model of $(\infty, n)$-category for general $n$.
In this talk, I will explore current work in the problem of taking homotopy bicategories of non-algebraic $(\infty, 2)$-categories, including a construction of my own. If time permits, I will discuss some of the applications of this problem to topological quantum field theories.
JS Lemay: Integrations in a Differential Category
Abstract: Numerous differential calculus related concepts have been formalized in a differential category. In particular, derivations and differential algebras from classical algebra generalize very nicely in a differential category. The integral analogues of these concepts are called integrations and Rota-Baxter algebras. In this talk, I will introduce the appropriate generalizations of integrations and Rota-Baxter algebras in a differential category.
Christian Williams: The Metalanguage of Category Theory
Abstract: Category theory is known as a language of mathematics. The fundamental concepts of the language are systematized in a fibrant double category, also known as a bicategory equipped with proarrows. We give a new definition of the structure: a bifibrant double category is a “two-sided bifibration” from a category to itself, with a weak composition and identity. We define the fully three-dimensional construction of pseudomonads, and thereby construct the metalanguage of all bifibrant double categories.
A category forms a (bi)fibrant double category, by forming the union of the arrow double category with its opposite; we call this the weave double category. Then a two-sided bifibration or matrix category is a span of categories forming a bimodule of weave double categories. We construct a three-dimensional category of categories, functors, profunctors, and matrix categories; squares are transformations, matrix functors, and matrix profunctors, and cubes are matrix transformations. This structure is a “bifibrant triple category without interchange”, which we call a metalogic.
A bifibrant double category is a pseudomonad in the metalogic of matrix categories. This defines the objects of a three-dimensional construction: a double functor is a morphism of pseudomonads, a vertical profunctor is a “vertical monad” between pseudomonads, and a horizontal profunctor is a bimodule of pseudomonads; a vertical transformation is a morphism of vertical monads, a horizontal transformation is a morphism of bimodules, and a double profunctor is a bimodule of vertical monads. A double transformation is a transformation of vertical bimodules. These form the metalogic of bifibrant double categories, whose “co/descent calculus” is a powerful unifying language of category theory.
Chris Grossack: 2-Categorical Descent and (Essentially) Algebraic Theories
Abstract: An essentially algebraic theory is an algebraic theory that moreover allows certain partially defined operations. Since algebraic theories enjoy certain nice properties that essentially algebraic theories don’t, it’s natural to ask if we can recognize when an essentially algebraic theory is actually algebraic. In the language of functorial semantics, this amounts to recognizing when a finite limit category is the free completion of a finite product category, and the problem can be solved by considering a 2-categorical descent theory. This was independent work, but writing it up I learned that the same result can already be found in a 1999 paper of Pedicchio and Wood. This seems to be less well known than it should be, and I hope this talk brings attention to this fascinating subject.
Nadja Egner: Internal double categories and 2-categories in the abelian context
Abstract: Given a category A, we can construct its arrow category Arr(A), whose objects are given by the morphisms in A and whose morphisms are given by commutative squares. If A is abelian, then Arr(A) is abelian and equivalent to the category Cat(A) of internal categories in A. We can repeat this construction and obtain Arr²(A), which is equivalent to the category Cat²(A) of internal double categories in A. The full subcategory 2-Arr(A) of Arr²(A), whose objects are given by pairs of composable morphisms whose composite is 0, is equivalent to the category 2-Cat(A) of internal 2-categories in A and reflective in Arr²(A). Moreover, it is closed under subobjects and quotients in Arr²(A) and a torsion-free subcategory thereof. We can characterize the (higher) extensions in Arr²(A) which are central with respect to 2-Arr(A) leading us to generalized Hopf formulae for homology. Regarding Arr²(A) as an instance of a category of diagrams in A and 2-Arr(A) as an instance of a full subcategory thereof, we obtain analogous results for the inclusions of n-Cat(A) into Catⁿ(A).
Elena Caviglia: What is a 2-stack?
Abstract is here.
Kristof Kanalas: Measuring how much a model is not positively closed
Abstract is here.
Amartya Shekhar: E_n-algebras in (m+1)-categories
Abstract: The fundamental question we address is: What’s an E_n-algebra in a (m+1)-category? We have coherence issues when we are trying to construct E_n, which are simplified by the Eckmann-Hilton Argument. The goal is to use a generalisation of the classical Eckmann-Hilton argument to get a minimal construction of E_n algebras. This is a joint work in progress with Yu Leon Liu.
Luca Mesiti: Grothendieck 2-topoi are elementary 2-topoi
Abstract:
A 2-dimensional generalization of subobject classifier has been proposed by Weber. It upgrades the classification process from pullbacks to comma objects, changing monomorphisms into discrete opfibrations. The archetypal example of elementary 2-topos is CAT , with the 2- classifier given by the construction of the category of elements. So that one could think of a 2-classifier as a Grothendieck construction inside a 2-category.
The conditions of a 2-classifier are hard to check in practice. Our main result is that these conditions can be checked just on a dense generator. The motivating examples are those of 2-dimensional presheaves and stacks, where representables form a dense generator. The classification process in 2-dimensional presheaves over representables is essentially Yoneda’s lemma, and our main result then provides a 2-classifier in 2-dimensional presheaves. This involves an indexed Grothendieck construction and a 2-dimensional notion of sieve.
We then restrict the 2-classifier in 2-dimensional presheaves to a 2-classifier in stacks. This is the main ingredient of a proof that Grothendieck 2-topoi are elementary 2-topoi.
Robin Cockett: The Granddad of all Computation
Schonfinkel invented “combinatory logic” — aka combinatory algebra (CA ) — in 1920. It consists of an application and two combinators S and K which satisfy just two identities. This simple gadget — amazingly — can generate all computable function.
In 1975 Solomon Feferman introduced partial combinatory algebra (PCA) and showed that it too generates all computable functions … but, notably, modelled the usual notion of computability given by Turing machines. A PCA consists of a partial binary operation and combinators S and K which satisfy just four identities.
In 2008 Pieter Hofstra and I introduced Turing Categories: we argued that this notion subsumed all the previous notions of computability. The initial Turing category — so the grandad of all computation — is generated by the generic PCA . While CAs are well known to have a confluent rewriting system — which is crucial as it is the rewriting system which implements computation — it still has not been documented in the literature that PCAs have a confluent rewriting system.
The talk will introduce Turing categories, PCAs, and discuss the rewriting system of PCAs.
richardblute.ca 
